432 RICE 



ART. K 



Gibbs and perhaps merit a brief remark here. When a body is 

 subject to a uniform stress in all directions we have 



Xx = Yy = Zz 

 and 



Xy — Yx = Yz = Zy = Zx = Xz = 0. 



If the body is isotropic, then referred to any axes 



en = 622 = 633 



and 



ei2 = 621 = 623 = 632 = 631 = 6i3 = 0. 



Thus along any line there is a fraction of elongation /, where 

 f = e — 1, e being the common value of en, 622, 633. Hence the 

 fraction of dilatation of volume is e* — 1 or practically 3/. 

 The quotient of the common value of Xy, Yy, Zz by 3/ is called 

 the bulk-modulus. (Gibbs calls it "elasticity of volume" on 

 page 213.) The conception is most important in the case of a 

 fluid. Here a variation of external thrust on the surface pro- 

 duces a variation of pressure from p to p -\- 8p; there results 

 from this an alteration of volume from v to v -{- 8v (8v is essen- 

 tially negative if 8p is essentially positive), i.e., a fraction of 



8v 

 dilatation 8v/v. The bulk-modulus is the limit of — 8p/— ; 



V 



i.e., it is 



dp(v, t) 



— V — - — ' 

 dv 



where p{v, t) is the function connecting pressure with volume 

 and temperature. (See [448].) This definition is synonymous 

 with the previous one, since for a liquid p = — Xx = —Yy = 

 —Zz and the shearing stresses vanish. (In fact the state of 

 stress uniform in all directions, mentioned above, is often 

 referred to as the case of "hydrostatic stress".) 



We can have a state of stress also in which the six constituents 



